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## What is the “real osculating space” of a (minimal) immersion?

In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else.

We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal immersion but I'm not sure it matters. It goes on to say

"Let $T_k(x)$ denote the real osculating space of order $k$ of $x$".

A) What is the precise definition of the real osculating space of an immersion in moden differential geometric language?

B) What does it mean intuitively?

[The paper is "An Extrinsic Rigidity Theorem for Minimal Immersions of S^2 into S^n" by J.L.M. Barbosa]

(I asked this on Stack Exchange originally)

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I took a quick look around and Wolfram Mathworld has a great animation of an osculating circle here. Also the entry on osculating curves here has a nice definition. The intuition is that when the you look at $S^{2}$ immersed in $S^{n}$ you take a point $x$ and ask for another surface which also contains $x$ and both the immersion of $S^{2}$ and this new surface have the same $k$ derivatives at $x$. So a tangent plane is an osculating surface of order 1.

However perhaps the best source of intuition is what the word osculate means in latin, to kiss.''

As for what the modern algebraic geometry term might be, I did some quick Googling for osculating algebraic geometry" and the term still seems to be in use. Perhaps not very commonly.

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View the immersion $x$ as an immersion into $R^{n+1}$. Then for each $p \in S^2$, there is a unique polynomial map $O_k: R^2 \rightarrow R^{n+1}$ of degree $k$ such that $O_k(0) = x(p)$ and the partials of $O_k$ of order $k$ or less at $0$ are equal to the corresponding covariant derivatives of $x$ at $p$.

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 Sorry - so do you know how the osculating space is then defined using this polynomial map? – Spencer Jan 25 at 10:38 It would be the image of the osculating map. – Deane Yang Jan 25 at 14:19